microeconomics 2. game theory - uni-bonn.de · pdf filemicroeconomics - 2.1 strategic form...

Microeconomics - 2.1 Strategic form games

Description idsds Nash Rationalisability Correlated eq

Microeconomics

2. Game Theory

Alex Gershkov

http://www.econ2.uni-bonn.de/gershkov/gershkov.htm

21. November 2007

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Strategic form games

1.a Describing a game in strategic form

1.b Iterated deletion of strictly dominated strategies

1.c Nash equilibrium and examples

1.d Mixed strategies

1.e Existence of Nash equilibria

1.f Rationilazability

1.g Correlated equilibrium

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Example: Entry game

1. 2 ice cream vendors decide whether or not to open an icecream stand in a particular street

2. the decision is taken without observing the other vendor’saction

3. if only one stand is opened, this vendor earns $ 3 (the othervendor earns zero)

4. if both stands open, they share profits and earn $ 1.5 each

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Let’s use a matrix to organise this information

entry no entryentry 1.5,1.5 3,0

no entry 0,3 0,0

we use the following conventions

1. rows contain vendor 1’s decisions, columns contain vendor 2’sdecisions

2. each outcome of the game is located in one cell of the matrix

3. outcomes (payoffs, profits) are given in expected utility in theform (u1, u2).

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This way of describing ‘simultaneous’ move games is calledstrategic form (sfg). Formally, a sfg consists of

1. a (not necessarily finite) set of players N: in the exampleN = 1, 2, generally N = 1, . . . , n

2. a (not necessarily finite) set of pure strategies S = S1 × S2: inthe example Si = entry , no entry, S = S2

i , i ∈ 1, 2,generally S = S1 × . . . × Sn = Sn

i

3. a set of payoff fns u(S): given as discrete values by thepayoffs in the sample matrix, generally by a vector of expectedutility fns u(S) = u1(S), . . . , un(S)

Thus a game in strategic form is fully described by N,S , u. A sfgis used to describe a game with no time dimension.

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Predicting the agent’s (=vendor’s) actions is easy provided that

1. payoffs (=profits) are the only thing agents care about

2. agents are rational

3. agents maximise their profits

Then the predicted actions are (entry,entry) leading to an outcomeof (1.5,1.5).

Notice that in this example an agent does not need to know theother agent’s choice in order to determine his optimal choice.

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1.b Equilibria (idsds)

Example:

l c rt 0,0 4,-1 1,-1

m -1,4 5,3 3,2b -1,2 0,1 4,1

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Definition: Given player i ’s pure strategy set Si , a mixed strategyfor player i , σi : Si −→ [0, 1] is a probability distribution over purestrategies. (Denote by Σi the space of player i mixed strategiesand Σ = Σ1 × . . . × Σn)

Another example:

l rt 2,0 -1,1

m 0,1 0,0b -1,0 2,2

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Definition: Pure strategy si is strictly dominated for player i ifthere exists σ′

i ∈ Σi such that

ui(σ′i , s−i ) > ui (si , s−i) for all s−i ∈ S−i (1)

where S−i = S1 × . . . × Si−1 × Si+1 × . . . × Sn

The strategy si is weakly dominated if there exists a σ′i ∈ Σi such

that inequality (1) holds weakly, that is, if the inequality is strictfor at least one s−i and equality holds for at least one s−i .

Definition: A rational player should never play a strictlydominated strategy.

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1.b Equilibria (idsds) & Rationality

Definition: X is called common knowledge (ck) between players Aand B if for any i ∈ A,B, i 6= j , k = 1, 2, 3, . . . KiKjKi . . .︸︷︷︸

k×

X

Idsds requires the following assumptions

1. players are rational

2. common knowledge of rationality

3. players know their payoffs

4. common knowledge of the structure of the game.

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DefinitionThe process of idsds proceeds as follows: Set S0

i = Si andΣ0

i = Σi . Define Ski recursively by

Ski = si ∈ Sk−1

i | there is no σi ∈ Σk−1i

s.t. ui (σi , s−i) > ui(si , s−i ) for all s−i ∈ Sk−1−i

and define

Σki = σi ∈ Σi |σi(si ) > 0 only if Si ∈ Sk

i

The set S∞i (Σ∞

i ) is the set of Pi ’s pure (mixed) strategies thatsurvive idsd.

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1.c Nash equilibrium

Game (‘Battle of the sexes’)

f of 1,2 0,0o 0,0 2,1

Here idsds does not work—we need a sharper tool: NE’qm.

NE’qm puts an additional restriction on the knowledge of playerscompared to idsds: It assumes that players have acorrect expectation of which prediction is played and subsequentlyhave no incentive to deviate from this prediction!

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Definition: A strategy profile s is a vector of dimension ncontaining a single strategy choice for each player.

Notation: s = (s1, . . . , sn) = (si , s−i)

Definition: Player i ’s best response Bi is the set of own strategiesgiving the highest payoff when the opponents play s−i . Formally

Bi(s−i ) = si ∈ Si |ui(si , s−i ) ≥ ui (s′i , s−i ), ∀s ′i ∈ Si

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Definition: A Nash equilibrium is a strategy profile s∗ whichprescribes a best response s∗i for each player i given that theopponents play s∗−i . Hence for every i ∈ N

ui(s∗i , s∗−i ) ≥ ui (s

′i , s

∗−i ), ∀s ′i ∈ Si

or equivalentlys∗i ∈ Bi(s

∗−i ), ∀i ∈ N.

Notice that according to this definition, the player’s expectedprediction of play is correct in equilibrium.

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1.c Nash equilibriumConsider the following game

l ru 3,2 2,0d 0,0 1,1

Observe thatB1(l) = u, B1(r) = u

whileB2(u) = l, B2(d) = r.

Hence the unique NE’qm is (u,l).

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1.d NE’qm & mixed strategiesWhat about this game, known as matching pennies?

h th 1,-1 -1,1t -1,1 1,-1

THIS GAME HAS NO PURE STRATEGY NE’QM!

Rem: A mixed strategy σi is a probability distribution over playeri ’s set of pure strategies Si denoted by Σi .

Assumption

We assume that i randomises independently from the otherplayers’ randomisations.

RemarkΣi is an |Si |-dimensional simplex Σi = p ∈ [0, 1]|Si | :

∑ph = 1.

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1.d NE’qm & mixed strategies

Definition: The mixed extension of the sfg N,S , u is the sfgN,Σ, u where i ′s payoff fn ui : ×j∈NΣj → R assigns to eachσ ∈ ×j∈NΣj the expected utility of the lottery over S induced by σ.

Definition: A mixed strategy NE’qm of a sfg is a NE’qm of itsmixed extension such that for every i ∈ N

σ∗i ∈ argmax

σi∈Σi

ui(σi , σ−i ).

LemmaIf σ∗ is a mixed strategy NE’qm then for every i ∈ N every pureaction si ∈ Si to which σ∗

i attaches positive probability is a bestresponse to σ∗

−i .

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It is sufficient to check that no player has a profitable pure-strategydeviation.

Alternative definition: A mixed strategy profile σ∗ is a NE’qm iffor all players i

ui (σ∗i , σ

∗−i ) ≥ ui (si , σ

∗−i) for all si ∈ Si .

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Theorem(Nash 1950) Every finite sfg N,S , u has a mixed strategy Nashequilibrium.

ui(σ∗i , σ

∗−i ) ≥ ui (σi , σ

∗−i ), ∀σi ∈ Σi .

Proof: The proof consists of showing that our setting satisfies allof Kakutani’s assumptions and then concluding that there exists aσ∗ which is contained in the set of best responses B(σ∗).

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1.e Existence of NE’qm

The existence question boils down to asking when the set ofintersections of best response correspondences

Bi(σ−i ) = σi ∈ Σi |ui (σi , σ−i ) ≥ ui(σ′i , σ−i ), ∀σ′

i ∈ Σi

is non-empty. The way Nash answered this question was to set upa mapping B : Σ → Σ where

B(σ) = (B1(σ−1)︸︷︷︸σ1

, . . . ,Bn(σ−n)︸︷︷︸σn

).

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DefinitionGiven X ⊂ R

n, a correspondence F between the sets X andY ⊂ R

k , written F : X Y , is a mapping that assigns a set F (x)to every x ∈ X .

DefinitionThe graph of a correspondence F : X Y is the setΓ(F ) = (x , y) ∈ X × Y | y ∈ F (x).

Hence the graph Γ(F ) is a set which can be checked for open orclosedness just like any other set.

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Theorem(Kakutani 1941)A correspondence F : X X satisfying

1. X is a convex, closed, and bounded subset of Rn,

2. F (·) is non-empty and convex-valued, and

3. F (·) has a closed graph

has a fixed point x∗ ∈ F (x∗) for x∗ ∈ X.

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1. Σi is a convex, closed, and bounded subset of Rn :

0.1 Σi is convex because if σi , σ′

i ∈ Σi then any mixtureλσi + (1 − λ)σ′

i ∈ Σi for all λ ∈ [0, 1].0.2 Σi is closed because its boundary points (assigning probability

0 and 1 to some pure strategy) are included in the set.0.3 Σi is bounded because all probabilities are bounded between

[0, 1].

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2. Bi(σ−i ) is non-empty and convex-valued:

0.1 There is a finite number of pure strategies (by assumption),therefore there exists at least one best reply to a given profile.Thus the best response correspondence cannot be empty.

0.2 Let σi , σ′

i ∈ Σi be best replies, then the mixture is a best replyas well because it will yield the same amount. Hence, Bi(σ−i )is convex valued.

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3. B(·) has a closed graph; thus any boundary point of the graphis included.

Assume the condition is violated, so there is a sequence(σk , σk) −→ (σ, σ) with σk ∈ B(σk), but σ /∈ B(σ). Then,σi /∈ Bi(σ−i ) for some player i . Thus, there is an ε > 0 and aσ′

i such that ui(σ′i , σ−i ) > ui(σi , σ−i ) + 3ε. Since ui is

continuous and (σk , σk) −→ (σ, σ), for k sufficiently large wehave

ui (σ′i , σ

k−i ) > ui(σ

′i , σ−i )−ε > ui(σi , σ−i )+2ε > ui(σ

ki , σk

−i )+ε.

Therefore, σ′i does strictly better against σk

−i than σki does,

contradiction to σk ∈ B(σk).

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1.f RationalisabilityBernheim’s (1984) and Pearce’s (1984) concept of Rationalisabilitystems from the desire to base an equilibrium concept on purelybehavioural assumptions (rationality)

agents view their opponents’ choice as an uncertain events every agent optimizes s.t. some probabilistic assessment of

uncertain event this assessment is consistent with all of his information the previous points are commonly known

The starting point is: What are all the strategies that a rationalplayer could play? A rational player will use only those strategiesthat are a best response to some belief he might have about thestrategies of his opponents.

DefinitionA player’s belief µi(σj ) is the conditional probability he attaches toopponent j playing a particular strategy σj . 26 / 41



1.f Rationalisability

Consider the following game due to Bernheim (1984)

b1 b2 b3 b4

a1 0,7 2,5 7,0 0,1a2 5,2 3,3 5,2 0,1a3 7,0 2,5 0,7 0,1a4 0,0 0,-2 0,0 10,-1

1. (a2, b2) is the only Nash equilibrium (and hence both a2 andb2 are rationalizable)

2. strategies a1, a3, b1 and b3 are rationalizable as well

3. a4 and b4 are not rationalizable

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DefinitionSet Σ0

i ≡ Σi and for each i recursively

Σki ≡ σi ∈ Σk−1

i |∃σ−i ∈ ×j 6=i Σk−1j

s.t. ui (σi , σ−i) ≥ ui(σ′i , σ−i ) for all σ′

i ∈ Σk−1i

The strategy σi of i is rationalizable if σi ∈⋂∞

k=0 Σki .

A strategy profile σ is rationalizable if σi is rationalizable for eachplayer i .

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1.f Rationalisability and idsd

Clearly, with any number of players, a strictly dominated strategyis never a best response: If σ′

i strictly dominates σi relative to Σ−i ,then σ′

i is a strictly better response than σi to every σ−i in Σ−i .

Thus, the set of rationalizable strategies is contained in the setthat survives iterated strict dominance.

Theorem(Pearce 1984) The set of rationalizable strategies and the set thatsurvives idsds coincide in two-players games.

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Proof: (Second direction) X - the set of strategies that surviveidsd.

We show that for every player every member of Xj is rationalizable.By definition, no action in Xj is strictly dominated in the gamewith the set of actions for each player i is Xi . ⇒ every action in Xj

is a BR among Xj to some beliefs on X−j .

We have to show that every action in Xj is a BR among Sj tosome beliefs on X−j . If sj ∈ Xj is not a BR among Sj then there isk s.t. sj is a BR among X k

j to a belief µj on X−j , but is not a BR

among X k−1j . Then, ∃bj ∈ X k−1

j \X kj that is a BR among X k−1

j toµj , contradiction, since bj was eliminated at n.

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1.h Correlated equilibrium

NE’qm assumes that players use independent randomisation inorder to arrive at their mutual actions. Given these, NE’qm is theminimal necessary condition for reasonable predictions (in generalgames). What if we dispense with this independence?

Consider that players can engage in (unmodelled)pre-play communication. Then they could conceivably decide tochoose their actions based on this communication or, moregenerally, on any other public (or private) event.

In order to capture this idea, let’s assume that the players canconstruct (or have access to) a signalling device which sends(public or private) signals on which the players can co-ordinatetheir actions.

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1.h Correlated equilibriumConsider the following example

l ru 5,1 0,0d 4,4 1,5

This game has 2 pureNE’qa (u, l), (d , r) plusthe mixed e’qm(( 1/2)u, ( 1/2)l) leading to(2.5, 2.5).

Can the players do any better?

1

1

2.5

2.5

5

5

u2(·)

u1(·)

(2.5,2.5)

(1,5)

(5,1)

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Suppose players can agree on constructing a public randomisationdevice with 2 equally likely outcomes H,T .

Assume that the publicly observable outcomes of therandomisations of this device are interpreted as the following planc :

if H (happening with pr(H) = 1/2), then play (u, l)

if T (happening with pr(T ) = 1/2), then play (d , r)

Then the expected payoffs from this plan c are

u1(c) =1

2u1(u, l) +

1

2u1(d , r) =

1

25 +

1

21 = 3 > 2.5!

(P2’s u2(·) is symmetric.)

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1.g Correlated equilibrium

In the same fashion, byusing their publicrandomisation device toco-ordinate their actions,the players can do betterthan Nash by attainingany point in the convexhull of NE’qa!

1

1

2.5

2.5

5

5

(3,3)

u2(·)

u1(·)

(2.5,2.5)

(1,5)

(5,1)

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Can we do still better?

Suppose that players can agree on establishing a device whichsends correlated but private signals to each player.

Consider a device with 3 equally likely outcomes A,B, C andassume the following information partitions:

P1’s partition P1 = A, B, C

P2’s partition P2 = A,B, C

Let’s look at the following plan d :

P1 plays u if told A and plays d if told B, C

P2 plays l if told A,B and plays r if told C

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Assumption

Players are Bayesian rational, ie. they use Bayes’ rule to updatetheir beliefs.

Claim: No player has an incentive to deviate from d .

Now P1’s overall payoff from using the plan d is

u1(d) =1

3u1(u, l) +

1

3u1(d , l) +

1

3u1(d , r) =

1

35 +

1

34 +

1

31 =

10

3> 3!

(Again the case for P2 is symmetric.)

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By using a privaterandomisation device,even payoffs outside theconvex hull of NE’qa areattainable!

1

1

2.5

2.5

5

5

(3,3)

u2(·)

u1(·)

(2.5,2.5)

(1,5)

(5,1)

(103

, 103

)

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DefinitionCommunication device is a triple (Ω,Hi , p) where

Ω is a state space (set of possible outcomes of the device)

p is a probability measure on Ω

Hi is information partition for i , if the true state is ω, Pi istold that the state lies in hi (ω), Pi ’s posterior about ω isgiven by Bayes’ rule p(ω|hi) = p(ω)/p(hi ) for ω ∈ hi and 0for ω /∈ hi

DefinitionA pure strategy for the expended game is ρi : Hi −→ Si

Alternatively,A pure strategy is ρi : Ω → Si , such that ρi (ω) = ρi (ω

′) wheneverω, ω′ ∈ hi for some hi ∈ Hi

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DefinitionA CE’qm ς relative to information structure (Ω,Hi , p) is a Nashequilibrium in strategies that are adapted to this informationstructure. That is. (ς1, ..., ςN ) is a correlated equilibrium if, forevery i and every adapted strategy ςi

∑

ω∈Ω

p(ω)ui(ςi (ω), ς−i (ω)) ≥∑

ω∈Ω

p(ω)ui (ςi (ω), ς−i (ω)).

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Theorem(Aumann 1974) For every finite game, the set of CE’qa isnon-empty.

Proof: Let the recommendation be an independent randomisationfor every player. Than the CE’qm is an independently mixedNE’qm which we know to exist. Hence the set of CE’qa containsthe set of NE’qa.

Theorem(Aumann 1974) The set of CE’qm payoffs of a sfg is convex.

Proof: One can always obtain a new CE’qm from publiclyrandomising between two other CE’qa.

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Since players haveaccess to a publicrandomisation device,the set of CE’qa isconvex (and containsthe set of NE’qa)!

1

1

2.5

2.5

5

5

(3,3)

u2(·)

u1(·)

(2.5,2.5)

(1,5)

(5,1)

(103

, 103

)

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microeconomics 2. game theory - uni-bonn.de · pdf filemicroeconomics - 2.1 strategic form...

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